Classification of Topological Insulators and Superconductors

Andreas W.W. LUDWIG, University of California, Santa Barbara

We present an exhaustive classification scheme of topological insulators and superconductors. The characteristic property of these unusual gapped phases of non-interacting Fermions is the appearance of topologically protected gapless extended degrees of freedom at the interface between a topologically trivial and a topologically non-trivial phase. In particular, these boundary degrees of freedom remain conducting even in the presence of strong disorder. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in dspatial dimensions to the problem of classifying Anderson localization at the (d-1)-dimensional boundary of the system. We find that in each spatial dimension there exist precisely five distinct classes of topological insulators (superconductors). The different topological sectors within a given such class can be labeled, depending on the case, by an integer winding number, or by a "binary" Z₂ quantity. One of the five classes of topological insulators is the "quantum spin Hall" (or: "Z₂-topological") insulator in d=2 and d=3 dimensions, experimentally observed in HgTe/(Hg,Ce)Te semiconductor quantum wells (d=2), and in BiSb alloys and Bi₂Se₃ (d=3). As we show, there are four additional classes of topological insulators (superconductors). For each spatial dimension d, the five classes of topological insulators are shown to correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than a decade ago by Altland and Zirnbauer in the context of disordered systems [generalizing the three well-known "Wigner-Dyson" symmetry classes (`unitary', 'orthogonal' and `symplectic')].

References

[1] A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Phys. Rev. B 78, 195125 (2008).

[2] A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, AIP Conf. Proc. 1134, 10 (2009). (Proceedings of the L.D. Landau Memorial Conference "Advances in Theoretical Physics").

[3] A.P. Schnyder, S. Ryu, A.W.W. Ludwig, Phys. Rev. Lett. 102, 196804 (2009).